The USA Today reports that the average expenditure on Valentine's Day is $100.89. Do male and female consumers differ in the amounts they spend? The average expenditure in a sample survey of 50 male consumers was $135.67, and the average expenditure in a sample survey of 38 female consumers was $68.64. Based on past surveys, the standard deviation for male consumers is assumed to be $40, and the standard deviation for female consumers is assumed to be $23. What is the point estimate of the difference between the population mean expenditure for males and the population mean expenditure for females (to 2 decimals)?

Answer:

Kay's husband drove at a speed of 50 mph

Step-by-step explanation:

This is a problem of simple motion.

First of all we must calculate how far Kay traveled to her job, and then estimate the speed with which her husband traveled later.

d=vt

v=45 mph

t= 20 minutes/60 min/hour = 0.333 h (to be consistent with the units)

d= 45mph*0.333h= 15 miles

If Kay took 20 minutes to get to work and her husband left home two minutes after her and they both arrived at the same time, it means he took 18 minutes to travel the same distance.

To calculate the speed with which Kate's husband made the tour, we will use the same initial formula and isolate the value of "V"

d=vt; so

v=[tex]\frac{d}{t}[/tex]

d= 15 miles

t= 18 minutes/60 min/hour = 0.30 h  (to be consistent with the units)

v=[tex]\frac{d}{t}=\frac{15 miles}{0.3 h}=50mph[/tex]

Kay's husband drove at a speed of 50 mph

Answer:

4-digit numbers with distinct digits and greater than 4500: 2800 numbers

5-digit numbers with distinct digits: 27216 numbers.

Step-by-step explanation:

If we represent a 4 number digit by ABCD, we have 9 posibilities for A (1,2,3,4,5,6,7,8 and 9, all but 0).

If every digit has to be different, we have 9 posibilities for B: ten digits (0,1,2,3,4,5,6,7,8 and 9 minus the one already used in A).

Int he same way, we have 8 posibilities for C and 7 for D.

Considering all 4-digits numbers, we have 9*9*8*7 = 4536 numers with distinct digits.

To know how many of these numbers are greater than 4500, we can substracte first the numbers that are smaller than 4000: A can take 3 digits (1,2 and 3) and B, C and D the same as before.

3*9*8*7 = 1512 numbers smaller than 4000

Then we can substrat the ones that are between 4000 and 4500

1*4*8*7 = 224 numbers between 4000 and 4500

So, if we substract from the total the numbers that are smaller than 4500 we have the results:

4-digit numbers with distinct digits greater than 4500 = 4536-(1512+224) = 2800

For 5-digit numbers, we can call the number ABCDE.

For A we have 9 digits possible (all but 0).

For B, we also have 9 posibilities (all digits but the one used in A).

For C, we have 8 digits (all 10 but the ones used in A and B).

For D, we have 7 digits.

For E, we have 6 digits.

Multiplying the possible combinations, we have:

9*9*8*7*6 =  27,216 5-digit numbers with distinct digits.

Two thousand five hundred twenty numbers have distinct digits and are more significant than 4500. There are 15120 5-digit odd numbers and 15120 5-digit even numbers with different numerals.

In Mathematics, to calculate how many numbers have distinct digits and are more significant than 4500, consider that any number greater than 4500 and less than 10000 is a 4-digit number. The thousands place can be filled by any number from 5 to 9, giving five options. Any ten digits can fill the hundreds place minus the one used in the thousands, giving nine options. The tens place, similarly, has eight votes. Similarly, the one's site has seven options, as the digit in that place cannot duplicate any of the prior digits. So, the answer is 5*9*8*7 = 2520 distinct numbers.

For the second part of your query, the 5-digit odd numbers with non-repeating digits, the tens place must be filled by five possible odd digits (1, 3, 5, 7, 9), and the first place can be filled by any number from 1 to 9, giving nine options. The other three areas have 8, 7, and 6 votes, respectively, leading to 9*8*7*6*5 = 15120 distinct numbers.

As for the 5-digit even numbers with non-repeating digits, the tens place can be filled by five possible even digits (0, 2, 4, 6, 8), and the first place can be filled by any number from 1 to 9, giving nine options. The other three areas have 8, 7, and 6 votes, respectively. So there are 9*8*7*6*5 = 15120 five-digit, distinct digit, even numbers.

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The actual movement of the boat is calculated by adding the vectors representing the velocity of the boat and the water current. The net vector shows both the speed and direction of the actual movement.

The actual movement of the boat is determined by adding vectorially, the velocity of the boat and the velocity of the current. This is because we need to consider both the speed and direction of the boat (traveling due east) and the current (flowing southwest).

Let's assume East as +i direction, and North as +j direction. So, the velocity of the boat is 22i km/hr and the velocity of the current is -10i/√2 -10j/√2 km/hr (as it is moving southwest).

To find the net velocity or actual movement, we add these two vectors.

Resultant velocity = 22i -10i/√2 -10j/√2 // Adding the i-components and the j-components

 = (22 -10/√2)i -10/√2 j km/hr

Therefore, the actual movement of the boat is (22 -10/√2)i -10/√2 j km/hr.

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Step-by-step explanation:

Say [tex]a[/tex] is an element of [tex]G[/tex] which might have more than 1 inverse. Let's call them [tex]b[/tex], and [tex]c[/tex]. So that [tex]a[/tex] has apparently two inverses, [tex]b[/tex] and [tex]c[/tex].

This means that [tex]a*b = e[/tex] and that [tex]a*c=e[/tex](where [tex]e[/tex] is the identity element of the group, and * is the operation of the group)

But so we could merge those two equations into a single one, getting

[tex]a*b=a*c[/tex]

And operating both sides by b by the left, we'd get:

[tex]b*(a*b)=b*(a*c)[/tex]

Now, remember the operation on any group is associative, meaning we can rearrange the parenthesis to our liking, gettting then:

[tex](b*a)*b=(b*a)*c[/tex]

And since b is the inverse of a, [tex]b*a=e[/tex], and so:

[tex](e)*b=(e)*c[/tex]

[tex]b=c[/tex] (since e is the identity of the group)

So turns out that b and c, which we thought might be two different inverses of a, HAVE to be the same element. Therefore every element of a group has a unique inverse.

Answer:

Water should be greater than 946.11 gallons per week to prevent the pool from drying.

Step-by-step explanation:

You have an outdoor swimming pool that is 5.0 m wide and 12.0 m long.

The height of water evaporated is = 2.35 inch

We will convert this to m.

1 inch = 0.0254 meter

So, 2.35 inch = [tex]2.35\times0.0254=0.05969[/tex] meter

Now, volume of water in the pool = [tex]5\times12\times0.05969=3.5814[/tex] cubic meter per week.

1 cubic meter = 264.172 gallons

So, 3.5814 cubic meter = [tex]3.5814\times264.172=946.11[/tex]gallons

Hence, the volume of the water that should be poured in the swimming pool should be greater than 946.11 gallons per week to prevent the pool from drying.

Answer and Explanation:

Given : Sides of right triangle 5,12 and 13.

To find : Show that the area of a right triangle of sides 5, 12 and 13 cannot be a square ?

Solution :

If 5,12 and 13 are sides of a right angle triangle then

13 is the hypotenuse as it is largest side.

then we take perpendicular as 12 and base as 5.

The area of the right angle triangle is

[tex]A=\frac{1}{2}\times b\times h[/tex]

Here, h=12 and b=5

[tex]A=\frac{1}{2}\times 5\times 12[/tex]

[tex]A=5\times 6[/tex]

[tex]A=30[/tex]

The area of the right angle triangle is 30 units.

30 is not a perfect square as [tex]30=2\times 3\times 5[/tex]

There is no square pair formed.

Answer:

In the first picture, correct answer is b: the inverse of function q is function r, because they are symmetrical about the line y = x.

In the second picture, correct answer is b: {y| 0<= y < 8}.

Step-by-step explanation:

The inverse function of f ( f⁻¹(x) ) must satisfy that: f(f⁻¹(x)) = f⁻¹(f(x)) = x; it returns every point x, transformed under function f, to its original place. In the graph, this property translates in the following statement: the inverse function of a function f is the reflection over the identity function (y = x).

In the graph shown in the question, the blue graph (r), is the one that corresponds to the reflection under the identity function. Therefore, the correct answer is b.

Regarding the second picture, first we need to understand what the range means. The range of a function corresponds to the set of all resulting values of the dependent variable. Since the values taken by the dependent variable span from 0 to 8 (including the 0 but not including the 8), then the answer is b: {y| 0<= y < 8}.

Answer:

40 bottles

Step-by-step explanation:

In the problem it is said that there were sold eigth times as much Cabernet Sauvignon as Zinfandel, this menas that if for example the cafe sold 1 bottle of zinfandel it would be 8 bottles of Cabernet, so we have to multiply the number of bottles sold fon Zinfandel by 8, as there were 5 bottles of sauvingon sold we have to multiply

8x5=40 bottles.

The Cabernet Cafe sold 40 bottles of Cabernet Sauvignon, given that it sold 8 times as many as the 5 bottles of Zinfandel.

The question involves a simple proportion calculation, typical of mathematics problems. Given the information that the Cabernet Cafe sold eight times as much Cabernet Sauvignon as Zinfandel, we can set up a ratio and solve for the unknown. In this case, we know that the number of Cabernet Sauvignon sold is 8 times the amount of Zinfandel sold. If the cafe sold 5 bottles of Zinfandel, using the said proportion, we multiply 5 (number of Zinfandel bottles sold) by 8 (the given ratio) to find the number of Cabernet Sauvignon bottles sold - which is 40 bottles.

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Answer:

Angle = 35°

Step-by-step explanation:

The figure would form a triangle and the tree will divide it into two 90 degree triangles.

Height of the triangles will be 8 feet which is opposite to the angle you have to find.

Hypotenuse of both 90 degree triangles will be 14 feet.

To find the angle, use the sin formula

sin (angle) = opposite/hypotenuse

angle = ?

opposite = 8

hypotenuse = 14

sin (angle) = 8/14

angle = sin inverse (0.57)

angle = 34.8° rounded off to 35°

The angle will be the same for both triangles.

Therefore, the acute angle each triangle makes with the ground is 35°.

Answer:

The nth term in the sequence is given by the equation:

[tex]n_{th} =(n+1)^{n-1}[/tex]

Step-by-step explanation:

Arranging a table for n and nth:

[tex]\left[\begin{array}{ccc}n&nth\\1&1\\2&3\\2&16\\4&125\\5&1296\end{array}\right][/tex]

It is easier to notice that 16 and 125 result from the second power of 4 and the third of 5, respectively, which are one number below their respective position. That is why we can deduce that the base of the power is n+1.

For n=2, the base n+1 results in 3, which matches the nth term for n=2. Since 3 is the result of 3 to the power of 1, 16 is 4 to the power of 2, and 125 is 5 to the power of 3, all the powers are one number behind n, so the power is given by n-1, giving the equation:

[tex]n_{th} =(n+1)^{n-1}[/tex]

We are 80% confident that the average net change in a student's score after completing the course lies between 7.88 and 21.12 points.

Here's how to construct the 80% confidence interval:

1. Calculate the sample mean and standard deviation:

Sample mean (xbar) = (23 + 18 + 23 + 12 + 13 + 23) / 6 = 18.33 points

Sample standard deviation (s) = √[(23 - 18.33)^2 + (18 - 18.33)^2 + ... + (23 - 18.33)^2] / (6 - 1) ≈ 5.38 points

2. Find the critical value for a 80% confidence interval:

For a two-tailed 80% confidence interval with 5 degrees of freedom (n-1), the critical value from the t-distribution table is approximately 1.942.

3. Calculate the margin of error:

Margin of error (ME) = critical value * standard deviation / √n

ME = 1.942 * 5.38 / √6 ≈ 4.49 points

4. Construct the confidence interval:

Lower limit = sample mean - margin of error = 18.33 - 4.49 ≈ 13.84 points

Upper limit = sample mean + margin of error = 18.33 + 4.49 ≈ 22.82 points

Therefore, we can be 80% confident that the true average net change in a student's score after completing the course falls within the range of 13.84 to 22.82 points, or rounded to the nearest whole number, 7.88 to 21.12 points.

Note: This calculation assumes the population is approximately normal. If there is reason to believe the population is not normal, a different method like bootstrapping may be more appropriate.

Answer: 532

Step-by-step explanation:

Let x be the number of students and y be the number of non-students.

Then,  by considering the given information, we have

[tex]x+y=817-----(1)\\\\2x+3y=1919-----------(2)[/tex]

Multiply 2 on the both sides of equation (1), we get

[tex]2x+2y=1634--------(3)[/tex]

Subtract (3) from (2), we get

[tex]y=285[/tex]

Put the value of y in (1), we get

[tex]x+285=817\\\\\Rightarrow\ x=817-285=532[/tex]

Hence, 532 student tickets were sold .

Answer:

less than

Step-by-step explanation:

When we are performing a two-tailed test of whether μ = 100,  the probability of detecting a shift of the mean to 105 will be ___Less__than___ the probability of detecting a shift of the mean to 110.

this means that

mean of 105 < mean of 110.

Answer:

Marginal revenue = R'(Q) = -0.6 Q + 221

Average revenue = -0.3 Q + 221

Step-by-step explanation:

As per the question,

Functions associated with the demand function P= -0.3 Q + 221, where Q is the demand.

Now,

As we know that the,

Marginal revenue is the derivative of the revenue function, R(x), which is equals the number of items sold,

Therefore,

R(Q) = Q × ( -0.3Q + 221) = -0.3 Q² + 221 Q

∴ Marginal revenue = R'(Q) = -0.6 Q + 221

Now,

Average revenue (AR) is defined as the ratio of the total revenue by the number of units sold that is revenue per unit of output sold.

[tex]Average\ revenue\ = \frac{Total\ revenue}{number\ of\ units\ sold}[/tex]

Where Total Revenue (TR) equals quantity of output multiplied by price per unit.

TR = Price (P) × Total output (Q) = (-0.3Q + 221) × Q = -0.3 Q² + 221 Q

[tex]Average\ revenue\ = \frac{TR}{Q}[/tex]

[tex]Average\ revenue\ = \frac{-0.3Q^{2}+221Q}{Q}[/tex]

∴ Average revenue = -0.3Q + 221

Answer:

  both domain and range are {-4, -3, -2, -1, 0, 1, 2, 3, 4}

  it IS a function

Step-by-step explanation:

The domain is the list of x-values of the plotted points. It is all the integers from -4 to +4, inclusive.

The range is the list of y-values of the plotted points. It is all the integers from -4 to +4, inclusive. (domain and range are the same for this function)

No x-value has more than one y-value associated with it, so this relation IS A FUNCTION.

I assume the first equation is supposed to be

[tex]5x-3y+2z=13[/tex]

and not

[tex]5x-3x+2x=4x=13[/tex]

As an augmented matrix, this system is given by

[tex]\left[\begin{array}{ccc|c}5&-3&2&13\\2&-1&-3&1\\4&-2&4&12\end{array}\right][/tex]

Multiply through row 3 by 1/2:

[tex]\left[\begin{array}{ccc|c}5&-3&2&13\\2&-1&-3&1\\2&-1&2&6\end{array}\right][/tex]

Add -1(row 2) to row 3:

[tex]\left[\begin{array}{ccc|c}5&-3&2&13\\2&-1&-3&1\\0&0&5&5\end{array}\right][/tex]

Multiply through row 3 by 1/5:

[tex]\left[\begin{array}{ccc|c}5&-3&2&13\\2&-1&-3&1\\0&0&1&1\end{array}\right][/tex]

Add -2(row 3) to row 1, and add 3(row 3) to row 2:

[tex]\left[\begin{array}{ccc|c}5&-3&0&11\\2&-1&0&4\\0&0&1&1\end{array}\right][/tex]

Add -3(row 2) to row 1:

[tex]\left[\begin{array}{ccc|c}-1&0&0&-1\\2&-1&0&4\\0&0&1&1\end{array}\right][/tex]

Multiply through row 1 by -1:

[tex]\left[\begin{array}{ccc|c}1&0&0&1\\2&-1&0&4\\0&0&1&1\end{array}\right][/tex]

Add -2(row 1) to row 2:

[tex]\left[\begin{array}{ccc|c}1&0&0&1\\0&-1&0&2\\0&0&1&1\end{array}\right][/tex]

Multipy through row 2 by -1:

[tex]\left[\begin{array}{ccc|c}1&0&0&1\\0&1&0&-2\\0&0&1&1\end{array}\right][/tex]

The solution to the system is then

[tex]\boxed{x=1,y=-2,z=1}[/tex]

Answer:

Everything is verified in the step-by-step explanation.

Step-by-step explanation:

We have the following differential equation:

[tex]x'' + 2x' + x = 0[/tex]

This differential equation has the following characteristic polynomial:

[tex]r^{2} + 2r + 1 = 0[/tex]

This polynomial has two repeated roots of [tex]r = -1[/tex].

Since the roots are repeated, our solution has the following format:

[tex]x(t) = c_{1}e^{-t} + c_{2}te^{-t}[/tex]

This shows that the family of solutions satisfies the equation for all values of the constants. The values of the constants depends on the initial conditions.

Lets solve the system with the initial conditions given in the exercise.

[tex]x(0) = 8[/tex]

[tex]c_{1}e^{0} + c_{2}(0)e^{0} = 8[/tex]

[tex]c_{1} = 8[/tex]

--------------------

[tex]x'(0) = 8[/tex]

[tex]x(t) = c_{1}e^{-t} + c_{2}te^{-t}[/tex]

[tex]x'(t) = -c_{1}e^{-t} + c_{2}e^{-t} - c_{2}te^{-t}[/tex]

[tex]-c_{1}e^{0} + c_{2}e^{0} - c_{2}(0)e^{0} = 8[/tex]

[tex]-c_{1} + c_{2} = 8[/tex]

[tex]c_{2} = 8 + c_{1}[/tex]

[tex]c_{2} = 8 + 8[/tex]

[tex]c_{2} = 16[/tex]

With these initial conditions, we have the following solution

[tex]x(t) = 8e^{-t} + 16te^{-t}[/tex]

Answer:

b. 5  

Step-by-step explanation:

In a stem and leaf plot,  each observation is split into a stem (the first digit or digits) and a leaf (usually the last digit).  

In this example, the entries in the 20s, 30s, and 40s are:

20, 20, 24, 25, 26, 35, 36, 43, 43

We omit 20, 20, 43, and 43 because they are not between 20 and 40.

That leaves the five observations

24, 25, 26, 35, 36

Answer:

The average cost per item when 200 items are produced is 61.6

Step-by-step explanation:

We start with the cost formula given by:

[tex]C(x)=7.6x+10,800[/tex]

Then we compute C(x) for x=200, 2000 and 5000 as follows:

[tex]C(200)=7.6*200+10,800=12,320\\C(2000)=7.6*2000+10,800=26,000\\C(5000)=7.6*5000+10,800=48,800[/tex]

Finally, to obtain the average cost per item when 200, 2,000 and 5,000 are produced (we will denote this by Av(200), Av(2000) and Av(5000) respectively) we just need to divide C(x) by the number of items produced. Then [tex]Av(x)=\frac{C(x)}{x}[/tex].

[tex]Av(200)=\frac{C(200)}{200}=\frac{12,320}{200}= 61.6\\Av(2000)=\frac{C(2000)}{2000}=\frac{26,000}{2000}= 13\\Av(5000)=\frac{C(5000)}{5000}=\frac{48,800}{5000}= 9.76\\[/tex]

A volt in terms of meters, seconds, kilograms, and coulombs is

[tex]$\frac{\text{kilogram} \times \text{meter}^2}{\text{second}^2 \times \text{coulomb}}$[/tex]

We are given that;

(vf)α=−2qαΔVmα

Now,

To find the equivalent of a volt in terms of basic SI units, we can use the definition of a volt as the potential difference that causes one joule of energy to be transferred per coulomb of charge.

A joule is the unit of energy, which is defined as the work done by a force of one newton over a distance of one meter.

A newton is the unit of force, which is defined as the product of mass and acceleration. Therefore, we can write:

[tex]$1 \text{ volt} = \frac{1 \text{ joule}}{1 \text{ coulomb}} = \frac{1 \text{ newton} \times 1 \text{ meter}}{1 \text{ coulomb}} = \frac{1 \text{ kilogram} \times 1 \text{ meter} \times 1 \text{ meter}}{1 \text{ second}^2 \times 1 \text{ coulomb}}$[/tex]

Simplifying, we get:

[tex]$1 \text{ volt} = \frac{\text{kilogram} \times \text{meter}^2}{\text{second}^2 \times \text{coulomb}}$[/tex]

Therefore, by volt answer will be [tex]$\frac{\text{kilogram} \times \text{meter}^2}{\text{second}^2 \times \text{coulomb}}$[/tex].

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A volt is a unit of potential difference, which is measured in joules per coulomb (V). 1 volt is equal to 1 joule per coulomb (1 V = 1 J/C).

The potential difference between two points A and B, VB - VA, is defined as the change in potential energy divided by the charge q. This potential difference is measured in joules per coulomb, which is called a volt (V). To express a volt in terms of the basic SI units of meters (m), seconds (s), kilograms (kg), and coulombs (C), we need to use the equation J = V × C, where J represents the unit of energy, the joule.

We can rewrite the equation J = V × C as V = J / C. Since 1 V is equivalent to 1 J/C, we can express the volt in terms of meters (m), seconds (s), kilograms (kg), and coulombs (C) as:

1 V = 1 J/C

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Answer:  a) 0.40    b) 0.15

Step-by-step explanation:

Let A denotes the event that students wear a watch and B denotes the event that students wear a bracelet.

Given : P(A)=0.25   ;   P(B)=0.30

[tex]P(A'\cup B')=0.60[/tex]

Since, [tex]P(A\cup B)=1-P(A'\cup B')=1-0.60=0.40[/tex]

Thus, the probability that this student is wearing a watch or a bracelet = 0.40

Also, [tex]P(A\cup B)=P(A)+P(B)-P(A\cap B)[/tex]

[tex]P(A\cap B)=P(A)+P(B)-P(A\cup B)[/tex]

[tex]P(A\cap B)=0.25+0.30-0.40\\\\\Rightarrow\ P(A\cap B)=0.15[/tex]

Thus,  the probability that this student is wearing both a watch and a bracelet= 0.15

Answer:

Step-by-step explanation:

Given that at a certain school, twenty-five percent of the students wear a watch and thirty percent wear a bracelet.

A- people who wear watch = 25%

B - people who wear bracelet = 30%

(AUB)' - People who wear neither a watch nor a bracelet=60%

[tex]A \bigcap B[/tex] - People who wear both =100%-60% = 40%

a) [tex]P(AUB) = P(A)+P(B)-P(AB) = 25%+30%-40%\\= 15%[/tex]

b) the probability that this student is wearing both a watch and a bracelet

= [tex]P(A \bigcap B) = 40%[/tex]

Answer:

Female

Step-by-step explanation:

The formula for finding z-score is;

z=(x-μ)/δ

where x= weight given=1600g

μ=mean weight given

δ=standard deviation given

For male

x=1600g

μ=3232.9g

δ=714.6g

z=(1600-3232.9)/714.6

z= -2.3

The weight for the male is 2.3 standard deviations below the mean

For female

x=1600g

μ=3094.9g

δ=586.3g

z=(1600-3094.9)/586.3

z= -2.5

The weight for the female is 2.5 standard deviations below the mean

⇒The female has the weight that is more extreme

Answer:

z= 1

Step-by-step explanation:

got it on egde

Answer: The ratio of boys to girls would be 3 : 5 .

Step-by-step explanation:

Since we have given that

Number of boys and girls = 32

Fraction of boys are going on a field trip = [tex]\dfrac{2}{3}[/tex]

Fraction of girls are going on a field trip = [tex]\dfrac{3}{4}[/tex]

Number of children left = 9

Let the number of boys be 'b'.

Let the number of girls be 'g'.

According to question, it becomes ,

[tex]b+g=32------------(1)\\\\32-9=\dfrac{2}{3}b+\dfrac{3}{4}g\\\\23=\dfrac{2b}{3}+\dfrac{3g}{4}-------------(2)[/tex]

From eq(1), we get that g = 32-b

So, it becomes,

[tex]\dfrac{2}{3}b+\dfrac{3}{4}(32-b)=23\\\\\dfrac{2}{3}b+24-\dfrac{3}{4}b=23\\\\\dfrac{2}{3}b-\dfrac{3}{4}b=23-24=-1\\\\\dfrac{8b-9b}{12}=-1\\\\\dfrac{-b}{12}=-1\\\\b=-1\times -12\\\\b=12[/tex]

so, number of girls would be 32 - b = 32 - 12 = 20

So, Ratio of boys to girls in class would be 12 : 20 = 3 : 5.

Therefore, the ratio of boys to girls would be 3 : 5 .

Answer:

The answer is : 0.0597

Step-by-step explanation:

Population mean = μ = 80

Standard deviation = σ = 20

Sample = N = 60

σ_mean = σ/√N

= (20)/√(60) = 2.582

Now we will find z1 and z2.

z1 = {(84) - μ}/σ_mean = [tex]{(84)-(80)}/(2.582)= 1.549[/tex]

z2 = {(88) - μ}/σ_mean = [tex]{(88)-(80)}/(2.582)= 3.098[/tex]

Now probability that the sample mean will be between 84 and 88 is given by:

Prob{ (1.549) ≤ Z≤ (3.098) } = (0.9990) -(0.9393) = 0.0597

[tex]\left[\begin{array}{ccc}a_{1}&b_{1}&c_{1}\\a_{2}&b_{2}&c_{2}\\a_{3}&b_{3}&c_{3}\end{array}\right]=\left[\begin{array}{ccc}-a_{1}^{2}-b_{1}^{2}\\-a_{2}^{2}-b_{2}^{2}\\-a_{3}^{2}-b_{3}^{2}\end{array}\right][/tex]

In the question,

We have to find out the circumcentre of the circle passing through the triangle with the vertices (a₁, b₁), (a₂, b₂) and (a₃, c₃).

So,

The circle is passing through these points the equation of the circle is given by,

[tex]x^{2}+y^{2}+ax+by+c=0[/tex]

On putting the points in the circle we get,

[tex]x^{2}+y^{2}+ax+by+c=0\\(a_{1})^{2}+(b_{1})^{2}+a(a_{1})+b(b_{1})+c=0\\and,\\(a_{2})^{2}+(b_{2})^{2}+a(a_{2})+b(b_{2})+c=0\\and,\\(a_{3})^{3}+(b_{3})^{3}+a(a_{3})+b(b_{3})+c=0\\[/tex]

So,

[tex](a_{1})^{2}+(b_{1})^{2}+a(a_{1})+b(b_{1})+c=0\\a(a_{1})+b(b_{1})+c=-(a_{1})^{2}-(b_{1})^{2}\,.........(1)\\and,\\a(a_{2})+b(b_{2})+c=-(a_{2})^{2}-(b_{2})^{2}\,.........(2)\\and,\\a(a_{3})+b(b_{3})+c=-(a_{3})^{3}-(b_{3})^{3}\,.........(3)\\[/tex]

On solving these equation using, Matrix method we can get the required equation of the circle,

[tex]\left[\begin{array}{ccc}a_{1}&b_{1}&c_{1}\\a_{2}&b_{2}&c_{2}\\a_{3}&b_{3}&c_{3}\end{array}\right]=\left[\begin{array}{ccc}-a_{1}^{2}-b_{1}^{2}\\-a_{2}^{2}-b_{2}^{2}\\-a_{3}^{2}-b_{3}^{2}\end{array}\right][/tex]

This is the required answer.

The center of the circumscribed circle of the symmetric isosceles triangle is at the origin, and the radius is equal to the length of the triangle's equal sides, denoted as r.

To find the center and the radius of the circle which circumscribes the triangle with vertices at ai, a, and a3, we must first understand the nature of the triangle. Given that the triangle is described to be symmetric with equal sides AB = BC = r, it is an isosceles triangle. The perpendicular bisector of the base a will pass through the midpoint of the base and the opposite vertex, given it is symmetric about this bisector. This will also be the diameter of the circumscribed circle. Consequently, as the triangle is isosceles and symmetric, we can use the properties of similar isosceles triangles to solve for the center and radius of the circumscribed circle.

Since the base a will also be the diameter of the circumscribed circle, and we know from geometry that the diameter is twice the radius (a = 2r), the radius of the circumscribing circle is r. The center of this circle is at the midpoint of the base a in the given symmetric form. Therefore, the center of the circle is at the origin due to the symmetric property and the radius remains r.

Answer:

214

Step-by-step explanation:

The playing field is 53.33 yards wide and 120 yards long you would need to find the area so multiply 53.33 by 120 yards. That equals 6399.6 , the thickness they are applying is 1.2 millimeters. You would divide the area, 6399.6 by 1.2 which would equal 5333. Divide that by 25 gallons and it equals 213.32, you would need to purchase 214, rounded.

Answer:

The number of containers to purchase is   [tex]N_V= 67.85[/tex]

Step-by-step explanation:

From the question we are told that

        The playing field  width is [tex]w_f = 53.33 \ yard = 53.33*0.9144 = 48.76m[/tex]

        The playing field length is [tex]l_f = 120 \ yards = 120 * 0.9144 = 109.728m[/tex]

The volume of one container is [tex]V= 25 \ gallon = 25 * 0.00378541 = 0.094625m^3[/tex]

        The thickness of the painting is  [tex]t = 1.2 \ mm = 1.2 * 0.001 = 0.0012m[/tex]

The area of the playing field is [tex]A = 48.76 * 109.728[/tex]

                                         [tex]=5350.337m^2[/tex]

The number of container of paint needed [tex]N_V[/tex] [tex]= \frac{area \ of \ playing \ field(A) * thickness \ of \ paint \ application(t) }{volume\ single \ container(V)}[/tex]

=>    [tex]N_V = \frac{5350.337 * 0.0012}{0.094625}[/tex]

             [tex]N_V= 67.85[/tex]

Answer:

x = 10

Step-by-step explanation:

8 + 12 = 2x

8 + 12 = 20

20 = 2x

----   -----

2      2

10 = x

x = 10

Hey!

----------------------------------------------------

Solution:

8 + 12 = 2x

~Divide 2 to both sides

20/2 = 2x/2

~Simplify

10 = x

----------------------------------------------------

Answer:

x = 10

----------------------------------------------------

Hope This Helped! Good Luck!

Answer:

  28.4 units

Step-by-step explanation:

If we call the given angles C and A, then the given side is c, and the other two sides can be found from the Law of Sines.

Angle B is the remaining angle of the triangle:

  180° -C -A = 180° -30° -45° = 105°

The remaining sides are ...

  b = sin(B)/sin(C)·c = sin(105°)/sin(30°)·6√2 ≈ 16.4

  a = sin(A)/sin(C)·c = sin(45°)/sin(30°)·6√2 = 12

Then the sum of the lengths of the remaining sides is ...

  a + b = 12 + 16.4 = 28.4 . . . units

By applying the principles of trigonometry and the Pythagorean theorem, we can determine that the sum of the lengths of the two remaining sides is approximately 25.5 units.

This question is about the application of trigonometric principles and the Pythagorean theorem. In a triangle, the sine of an angle is defined as the ratio of the side opposite that angle to the hypotenuse. Given a 30-degree angle and its opposite side of length 6√2, we can find the hypotenuse (h) using the fact that sin(30) = 1/2. So, 1/2 = 6√2 / h. Solving this, we get h = 2 * 6√2 = 12√2.

The angle 45 degrees helps determine the length of the remaining side. Using the fact that cos(45) = s/h, where s is the remaining side, we get cos(45) = s / 12√2. Solving for s, s = cos(45) * 12√2, s = √2/2 * 12√2 = 6√2.

So, the sum of the lengths of the two remaining sides is 12√2 + 6√2 = 18√2, which is approximately 25.5 when rounded to the nearest tenth.

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A.

Let Annie's weight be = a

Let Benjie's weighs = b

Let Carmen's weight be = c

One day Annie weighed 24 ounces more than Benjie, equation forms:

[tex]a=b+24[/tex]        ......(1)

Benjie weighed 3 1/4 pounds less than Carmen.  

In ounces:

1 pound = 16 ounces

[tex]\frac{13}{4}[/tex] pounds = [tex]\frac{13}{4}\times16=52[/tex] ounces

[tex]b=c-52[/tex]  or

[tex]c=b+52[/tex]    ......(2)

Now adding (1) and (2), we get

a+b=b+24+c-52

=> [tex]a=c-28[/tex]

This gives  Annie weighs 28 ounces less than Carmen.

B.

We cannot know anyone's actual weight, as we only know their relative weights.

Answer:

There is a 0.64% probability that the costumer has filed a claim.

Step-by-step explanation:

Probability:

What you want to happen is the desired outcome.

Everything that can happen iis the total outcomes.

The probability is the division of the number of possible outcomes by the number of total outcomes.

Our problem has these following probabilities:

-78% that a costumer is a good risk.

-20% that a costumer is a medium risk.

-2% that a costumer is a poor risk.

Also:

- 0.5% of a good risk costumer filling an accident claim

- 1% of a medium risk costumer filling an accident claim.

-2.5% of a poor risk costumer filling an accident claim.

The question is:

What is the probability that the customer has filed a claim?

[tex]P = P_[1} + P_{2} + P_{3}[/tex], in which:

-[tex]P_{1}[/tex] is the probability that a good risk costumer is chosen and files a claim. This probability is: the probability of a good risk costumer being chosen multiplied by the probability that a good risk costumer files a claim. So:

[tex]P_[1} = 0.78*0.005 = 0.0039[/tex]

-[tex]P_{2}[/tex] is the probability that a medium risk costumer is chosen and files a claim. This probability is: the probability of a medium risk costumer being chosen multiplied by the probability that a medium risk costumer files a claim. So:

[tex]P_[2} = 0.20*0.01 = 0.002[/tex]

-[tex]P_{3}[/tex] is the probability that a poor risk costumer is chosen and files a claim. This probability is: the probability of a poor risk costumer being chosen multiplied by the probability that a poor risk costumer files a claim. So:

[tex]P_[3} = 0.02*0.025 = 0.0005[/tex]

[tex]P = P_[1} + P_{2} + P_{3} = 0.0039 + 0.002 + 0.0005 = 0.0064[/tex]

There is a 0.64% probability that the costumer has filed a claim.

The probability that a randomly selected customer has filed a claim is calculated using the total law of probability, which yields a result of 0.0064 or 0.64%, after considering the probabilities of each risk group filing a claim.

To calculate the probability that a randomly chosen customer has filed a claim, we need to use the total law of probability. This involves multiplying the probability of a customer being in each risk group by the probability that a customer in that risk group files a claim, and then summing these products.

For good risks, this probability is 0.78 (the percentage of good risk customers) multiplied by 0.005 (the probability a good risk customer files a claim): 0.78 * 0.005 = 0.0039.

For medium risks, the calculation is 0.20 * 0.01 = 0.0020.

For poor risks, the calculation is 0.02 * 0.025 = 0.0005.

Adding these probabilities together gives us the total probability that a customer has filed a claim: 0.0039 + 0.0020 + 0.0005 = 0.0064, or 0.64% if expressed as a percentage.

Therefore, the probability that a randomly selected customer from the insurance company has filed a claim is 0.0064 or 0.64%, rounded to four decimal places.